GK dimension of some connected Hopf algebras

Abstract

While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite, we have yet to determine the Gelfand–Kirillov (GK) dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in Wang, D. G., Zhang, J. J., Zhuang, G. (2013). Coassociative lie algebras. Glasgow Math. J. 55(A):195–215 to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the GK dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements.

Keywords:

• Gelfand–Kirillov dimension
• Hopf algebras

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16T05

Acknowledgements

The author would like to thank his adviser Allen Bell, and the Mathematics Department at both University of Wisconsin-Milwaukee and Bradley University for their guidance and support of the author’s work. The author is also grateful to the referees for their constructive feedback and advice.